A363616 -id:A363616 - cp's OEIS frontend

A325940 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k)^2.

0, 1, -2, 4, -4, 4, -6, 11, -10, 6, -10, 18, -12, 8, -20, 26, -16, 13, -18, 28, -28, 12, -22, 48, -28, 14, -36, 38, -28, 24, -30, 57, -44, 18, -44, 62, -36, 20, -52, 74, -40, 32, -42, 58, -72, 24, -46, 110, -54, 31, -68, 68, -52, 40, -68, 100, -76, 30, -58, 116

Offset: 1

Ilya Gutkovskiy, Sep 09 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A002129, A048272, A065608, A128315, A363615, A363616.

A325940:= func< n | (&+[0^(n mod j)*(-1)^j*(j-1): j in [1..n]]) >;
[A325940(n): n in [1..70]]; // G. C. Greubel, Jun 22 2024

nmax = 60; CoefficientList[Series[Sum[x^(2 k)/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^d (d - 1), {d, Divisors[n]}], {n, 1, 60}]

{a(n) = sumdiv(n, d, (-1)^d*(d-1))} \\ Seiichi Manyama, Sep 14 2019

def A325940(n): return sum(0^(n%j)*(-1)^j*(j-1) for j in range(1, n+1))
[A325940(n) for n in range(1,71)] # G. C. Greubel, Jun 22 2024

G.f.: Sum_{k>=2} (-1)^k * (k - 1) * x^k / (1 - x^k).
a(n) = Sum_{d|n} (-1)^d * (d - 1).
a(n) = A048272(n) - A002129(n).
Faster converging series: A(q) = Sum_{n >= 1} (-1)^n*q^(n^2)*((n-1)*q^(3*n) + n*q^(2*n) + (n-2)*q^n + n-1)/((1 + q^n)*(1 - q^(2*n))) - apply the operator t*d/dt to equation 1 in Arndt, then set t = -q and x = q. - Peter Bala, Jan 22 2021
a(n) = A128315(n, 2). - G. C. Greubel, Jun 22 2024

A128315 Inverse Moebius transform of signed A007318.

Original entry on oeis.org

1, 0, 1, 2, -2, 1, -1, 4, -3, 1, 2, -4, 6, -4, 1, 0, 4, -9, 10, -5, 1, 2, -6, 15, -20, 15, -6, 1, -2, 11, -24, 36, -35, 21, -7, 1, 3, -10, 29, -56, 70, -56, 28, -8, 1, 0, 6, -30, 80, -125, 126, -84, 36, -9, 1, 2, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1, -2, 18, -67, 176, -335, 463, -462, 330, -165, 55, -11, 1

Offset: 1

Gary W. Adamson, Feb 25 2007

A128316 = A000012 * A128315.

			First few rows of the triangle:
   1;
   0,  1;
   2, -2,  1;
  -1,  4, -3,  1;
   2, -4,  6, -4,  1;
   0,  4, -9, 10, -5, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000034, A000984, A007318, A010693, A048272, A051731, A128316.
Cf. A130595, A325940, A363615, A363616.
Cf. A000012 (row sums).

A128315:= func< n,k | (&+[0^(n mod j)*(-1)^(k+j)*Binomial(j-1, k-1): j in [k..n]]) >;
[A128315(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 22 2024

A128315[n_, k_]:= (-1)^k*DivisorSum[n, (-1)^#*Binomial[#-1, k-1] &];
Table[A128315[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 22 2024 *)

def A128315(n,k): return sum( 0^(n%j)*(-1)^(k+j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128315(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 22 2024

T(n, k) = A051731(n, k) * A130595(n-1, k-1) as infinite lower triangular matrices.
T(n, 1) = A048272(n).
Sum_{k=1..n} T(n, k) = A000012(n) = 1 (row sums).
From G. C. Greubel, Jun 22 2024: (Start)
T(n, k) = (-1)^k * Sum_{d|n} (-1)^d * binomial(d-1, k-1).
T(n, 2) = A325940(n), n >= 2.
T(n, 3) = A363615(n), n >= 3.
T(n, 4) = A363616(n), n >= 4.
T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A081295(n).
Sum_{k=1..n} k*T(n, k) = A000034(n-1), n >= 1.
Sum_{k=1..n} (k+1)*T(n, k) = A010693(n-1), n >= 1. (End)

a(43) = 28 inserted and more terms from Georg Fischer, Jun 05 2023

A363615 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^3.

Original entry on oeis.org

0, 0, 1, -3, 6, -9, 15, -24, 29, -30, 45, -67, 66, -63, 98, -129, 120, -117, 153, -204, 206, -165, 231, -341, 282, -234, 354, -417, 378, -354, 435, -594, 542, -408, 582, -770, 630, -513, 770, -966, 780, -702, 861, -1071, 1072, -759, 1035, -1527, 1143, -930, 1346

Offset: 1

Seiichi Manyama, Jun 11 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002129, A048272, A128315, A325940, A321543, A363610, A363616.

A363615:= func< n | -(&+[(-1)^d*Binomial(d-1,2): d in Divisors(n)]) >;
[A363615(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024

a[n_] := -DivisorSum[n, (-1)^#*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)

my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1+x^k)^3)))

a(n) = -sumdiv(n, d, (-1)^d*binomial(d-1, 2));

def A363615(n): return sum(0^(n%j)*(-1)^(j+1)*binomial(j-1,2) for j in range(1, n+1))
[A363615(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

G.f.: -Sum_{k>0} binomial(k-1,2) * (-x)^k/(1 - x^k).
a(n) = -Sum_{d|n} (-1)^d * binomial(d-1,2).
a(n) = A128315(n, 3), for n >= 3. - G. C. Greubel, Jun 22 2024
a(n) = (A321543(n) - 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025

A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

Original entry on oeis.org

1, -3, 11, -23, 36, -49, 85, -143, 176, -188, 287, -433, 456, -479, 726, -959, 970, -1024, 1331, -1748, 1866, -1741, 2301, -3153, 2961, -2824, 3830, -4559, 4496, -4514, 5457, -6943, 6842, -6174, 7890, -9844, 9140, -8553, 11126, -13348, 12342, -11998, 14191, -16941

Offset: 1

Ilya Gutkovskiy, Oct 23 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A000292, A000593, A001157, A001158, A002129, A050999, A051000, A059358, A138503, A320900, A321543.

Cf. A363598, A363616, A363617, A363631.

seq(coeff(series(add(x^k/(1+x^k)^4,k=1..n),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 23 2018

nmax = 44; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 44}]

a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)*(d + 2)/6); \\ Amiram Eldar, Jan 04 2025

G.f.: Sum_{k>=1} (-1)^(k+1)*A000292(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)*(d + 2)/6.
a(n) = (4*A000593(n) + 6*A050999(n) + 2*A051000(n) - 2*A000203(n) - 3*A001157(n) - A001158(n))/6.
a(n) = (A138503(n) + 3*A321543(n) + 2*A002129(n)) / 6. - Amiram Eldar, Jan 04 2025

A363631 Expansion of Sum_{k>0} (1/(1+x^k)^4 - 1).

Original entry on oeis.org

-4, 6, -24, 41, -60, 70, -124, 206, -244, 236, -368, 560, -564, 566, -896, 1175, -1144, 1180, -1544, 2042, -2168, 1942, -2604, 3650, -3336, 3100, -4304, 5096, -4964, 4940, -5988, 7720, -7528, 6636, -8616, 10809, -9884, 9126, -12064, 14548, -13248, 12796, -15184, 18192, -18412, 15830

Offset: 1

Seiichi Manyama, Jun 12 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363629, A363630.

Cf. A320901, A363598, A363616, A363617.

a[n_] := DivisorSum[n, (-1)^#*Binomial[# + 3, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)

a(n) = sumdiv(n, d, (-1)^d*binomial(d+3, 3));

G.f.: Sum_{k>0} binomial(k+3,3) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d+3,3).

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A325940 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k)^2.

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A128315 Inverse Moebius transform of signed A007318.

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A363615 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^3.

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A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

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A363631 Expansion of Sum_{k>0} (1/(1+x^k)^4 - 1).

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